Abstract
We study the class of -harmonic -quasiconformal mappings with angular ranges. After building a differential equation for the hyperbolic metric of an angular range, we obtain the sharp bounds of their hyperbolically partial derivatives, determined by the quasiconformal constant . As an application we get their hyperbolically bi-Lipschitz continuity and their sharp hyperbolically bi-Lipschitz coefficients.
Highlights
Let Ω and Ω be two domains of hyperbolic type in the complex plane C
A euclidean harmonic mapping defined on a connected domain is of the form f h g, where h and g are two analytic functions in Ω
In this paper we study the class of 1/|w|2-harmonic mappings
Summary
Let Ω and Ω be two domains of hyperbolic type in the complex plane C. In this paper we study the corresponding question for the class of 1/|w|2-harmonic quasiconformal mappings To this question, Examples 5.1, and 5.2 show that if the metric ρ is not necessary to be smooth in the range of a ρ-harmonic quasiconformal mapping f, f generally does not need to have euclidean and hyperbolically Lipschitz continuity even if its range is convex. In this paper we will study the hyperbolically Lipschitz or bi-lipschitz continuity of a 1/|w|2-harmonic quasiconformal mapping with an angular range and its sharp hyperbolically Lipschitz coefficient determined by the constant of quasiconformality. The key of this paper is to build a differential equation for the hyperbolic metric of an angular domain, which is different for using a differential inequality when we studied the class of euclidean harmonic quasiconformal mappings in 14. In order to show the sharpness of Theorems 3.1 and 4.1, we present two examples satisfying that the inequalities 3.2 no longer hold for two classes of 1/|w|2-harmonic quasiconformal mappings with nonangular ranges see Examples 5.4 and 5.5
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Mathematics and Mathematical Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
